Subgroups — C₂×#22609305 #33068232

Order 18688.

758 conjugacy classes of subgroups.

The subgroup lattice has 388 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#17988983	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#17988878	C₂	1	1	yes	maximal, characteristic
3	C₂×#17988789	C₂	1	1	yes	maximal, characteristic
4	C₂×#17988652	C₂	1	1	yes	maximal, characteristic
5	#22609305	C₂	1	8	yes	maximal, direct product factor
6	C₂×#17988337	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×#17988335	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#332×C₇₃	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂²×#11924015	—	2	1	no	—
10	C₂²×C₈×D₇₃	—	2	1	no	—
11	C₂×#14556591	C₂²	1	1	yes	characteristic
12	C₂×C₄×D₂₉₂	C₂²	1	1	yes	characteristic
13	#17988983	—	2	2	no	—
14	#17988983	C₂²	1	4	yes	complemented
15	#17988878	—	2	2	no	—
16	#17988878	C₂²	1	4	yes	—
17	#17988789	—	2	2	no	—
18	#17988789	C₂²	1	4	yes	—
19	#17988652	—	2	2	no	—
20	#17988652	C₂²	1	4	yes	—
21	C₂×#14556523	—	2	1	no	—
22	C₂×#14556522	—	2	1	no	—
23	C₂×#14556522	—	2	1	no	—
24	C₂×C₄⋊#10596831	C₂²	1	1	yes	characteristic
25	C₂×#14556515	—	2	1	no	—
26	C₂×#14556513	—	2	1	no	—
27	C₂×#14556503	—	2	1	no	—
28	C₂×C₄×#10596831	C₂²	1	1	yes	characteristic
29	#17988337	C₂²	1	4	yes	complemented
30	#17988335	C₂²	1	4	yes	complemented
31	C₂×C₄⋊Q₈×C₇₃	C₂²	1	1	yes	characteristic
32	C₂×C₄⋊C₈×C₇₃	C₂²	1	1	yes	characteristic
33	C₂×C₄⋊C₈×C₇₃	C₂²	1	1	yes	characteristic
34	#332×C₇₃	C₂²	1	4	yes	complemented
35	C₂²×Dic₁₄₆	—	2	1	no	—
36	C₂²×D₂₉₂	D₄	1	1	yes	characteristic
37	C₂²×C₄×D₇₃	—	2	1	no	—
38	C₂×#11924015	—	2	4	no	—
39	C₂×#11924015	—	4	4	no	—
40	C₂²×#10596831	—	2	1	no	—
41	C₂²×#10596831	D₄	1	1	yes	characteristic
42	C₂²×#10596831	D₄	1	1	yes	characteristic
43	C₂×C₈×D₇₃	—	2	4	no	—
44	C₂×C₈×D₇₃	—	4	4	no	—
45	#14556591	—	2	1	no	—
46	#14556591	—	2	1	no	—
47	#14556591	—	2	1	no	—
48	#14556591	C₂³	1	2	yes	—
49	C₄×D₂₉₂	—	2	1	no	—
50	C₄×D₂₉₂	—	2	1	no	—
51	C₄×D₂₉₂	D₄	1	2	yes	—
52	C₄×D₂₉₂	C₂³	1	2	yes	—
53	C₂×C₄⋊Dic₇₃	—	2	1	no	—
54	C₂×C₄⋊Dic₇₃	—	2	1	no	—
55	C₂×C₄⋊Dic₇₃	D₄	1	1	yes	characteristic
56	C₂×#11924005	—	2	1	no	—
57	#14556523	—	2	4	no	—
58	#14556522	—	2	4	no	—
59	#14556522	—	2	4	no	—
60	C₄⋊#10596831	—	2	1	no	—
61	C₄⋊#10596831	C₂³	1	2	yes	—
62	#14556515	—	2	4	no	—
63	#14556513	—	2	4	no	—
64	#14556503	—	2	4	no	—
65	C₄×#10596831	D₄	1	2	yes	—
66	C₄×#10596831	C₂³	1	2	yes	—
67	C₂²×Q₈×C₇₃	—	2	1	no	—
68	C₂²×C₈×C₇₃	—	2	1	no	—
69	C₂²×C₈×C₇₃	—	2	1	no	—
70	C₄⋊Q₈×C₇₃	—	2	1	no	—
71	C₄⋊Q₈×C₇₃	—	2	1	no	—
72	C₄⋊Q₈×C₇₃	—	2	1	no	—
73	C₄⋊Q₈×C₇₃	C₂³	1	2	yes	—
74	C₂×C₄⋊C₄×C₇₃	—	2	1	no	—
75	C₂×C₄⋊C₄×C₇₃	—	2	1	no	—
76	C₂×C₄²×C₇₃	C₂³	1	1	yes	characteristic
77	C₄⋊C₈×C₇₃	—	2	1	no	—
78	C₄⋊C₈×C₇₃	—	2	1	no	—
79	C₄⋊C₈×C₇₃	C₂³	1	2	yes	—
80	C₄⋊C₈×C₇₃	C₂³	1	2	yes	—
81	C₂³×D₇₃	—	2	1	no	—
82	C₂×Dic₁₄₆	—	2	2	no	—
83	C₂×Dic₁₄₆	—	4	1	no	—
84	C₂×Dic₁₄₆	—	8	2	no	—
85	C₂×D₂₉₂	—	2	1	no	—
86	C₂×D₂₉₂	—	2	2	no	—
87	C₂×D₂₉₂	SD₁₆	1	4	yes	—
88	C₂×D₂₉₂	C₂×D₄	1	2	yes	—
89	C₂²×Dic₇₃	—	2	1	no	—
90	C₂²×Dic₇₃	—	4	1	no	—
91	C₂×C₄×D₇₃	—	2	2	no	—
92	C₂×C₄×D₇₃	—	2	2	no	—
93	C₂×C₄×D₇₃	—	4	2	no	—
94	C₂×C₄×D₇₃	—	4	2	no	—
95	#11924015	—	4	8	no	—
96	C₂×#10596831	—	2	2	no	—
97	C₂×#10596831	—	2	2	no	—
98	C₂×#10596831	—	4	2	no	—
99	C₂×#10596831	SD₁₆	1	4	yes	complemented
100	C₂×#10596831	C₂×D₄	1	2	yes	—
101	C₂×#10596831	C₂×D₄	1	2	yes	—
102	C₈×D₇₃	—	4	8	no	—
103	C₄⋊Dic₇₃	—	2	1	no	—
104	C₄⋊Dic₇₃	—	2	2	no	—
105	C₄⋊Dic₇₃	—	2	2	no	—
106	C₄⋊Dic₇₃	—	4	1	no	—
107	C₄⋊Dic₇₃	—	4	1	no	—
108	C₄⋊Dic₇₃	C₂×D₄	1	2	yes	—
109	#11924005	—	2	2	no	—
110	#11924005	—	2	2	no	—
111	C₂×Q₈×C₇₃	—	2	2	no	—
112	C₂×Q₈×C₇₃	—	4	1	no	—
113	C₂×Q₈×C₇₃	—	4	2	no	—
114	C₂×Q₈×C₇₃	—	4	2	no	—
115	C₂²×C₄×C₇₃	—	4	1	no	—
116	C₂²×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
117	C₂²×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
118	C₂²×C₄×C₇₃	#41	1	1	yes	characteristic
119	C₂×C₈×C₇₃	—	2	2	no	—
120	C₂×C₈×C₇₃	—	2	2	no	—
121	C₂×C₈×C₇₃	—	4	2	no	—
122	C₂×C₈×C₇₃	—	4	2	no	—
123	C₄⋊C₄×C₇₃	—	2	2	no	—
124	C₄⋊C₄×C₇₃	—	2	2	no	—
125	C₄⋊C₄×C₇₃	—	4	1	no	—
126	C₄⋊C₄×C₇₃	—	4	1	no	—
127	C₄²×C₇₃	C₂×D₄	1	1	yes	characteristic
128	C₄²×C₇₃	C₂×D₄	1	1	yes	characteristic
129	C₄²×C₇₃	#41	1	1	yes	characteristic
130	C₄²×C₇₃	C₂⁴	1	1	yes	characteristic
131	C₂³×C₇₃	#121	1	1	yes	characteristic
132	C₂²×D₇₃	—	2	2	no	—
133	C₂²×D₇₃	—	2	4	no	—
134	C₂²×D₇₃	—	4	2	no	—
135	C₂²×D₇₃	—	4	2	no	—
136	Q₈×C₇₃	—	4	2	no	—
137	Q₈×C₇₃	—	4	2	no	—
138	Q₈×C₇₃	—	8	2	no	—
139	C₂×C₄×C₇₃	—	2	2	no	—
140	C₂×C₄×C₇₃	—	4	2	no	—
141	C₂×C₄×C₇₃	—	8	2	no	—
142	C₂×C₄×C₇₃	#121	1	1	yes	characteristic
143	C₂×C₄×C₇₃	#121	1	1	yes	characteristic
144	C₂×C₄×C₇₃	#121	1	1	yes	characteristic
145	C₂×C₄×C₇₃	C₂×SD₁₆	1	2	yes	—
146	C₂×C₄×C₇₃	C₂×SD₁₆	1	2	yes	—
147	C₂×C₄×C₇₃	#135	1	2	yes	—
148	C₂×C₄×C₇₃	#137	1	2	yes	—
149	C₂×C₄×C₇₃	C₂²×D₄	1	1	yes	characteristic
150	C₂×C₄×C₇₃	C₂²×D₄	1	1	yes	characteristic
151	C₂×C₄×C₇₃	C₂×#41	1	1	yes	characteristic
152	C₂×Dic₇₃	—	2	2	no	—
153	C₂×Dic₇₃	—	4	2	no	—
154	C₂×Dic₇₃	—	4	2	no	—
155	C₂×Dic₇₃	—	8	2	no	—
156	D₂₉₂	—	2	2	no	—
157	D₂₉₂	—	2	4	no	—
158	D₂₉₂	C₂×SD₁₆	1	4	yes	—
159	C₄×D₇₃	—	4	2	no	—
160	C₄×D₇₃	—	4	2	no	—
161	C₄×D₇₃	—	4	4	no	—
162	Dic₁₄₆	—	8	2	no	—
163	Dic₁₄₆	—	8	2	no	—
164	C₈×C₇₃	—	4	2	no	—
165	C₈×C₇₃	—	4	2	no	—
166	#10596831	—	2	2	no	—
167	#10596831	—	4	2	no	—
168	#10596831	C₂×SD₁₆	1	4	yes	complemented
169	C₂²×C₇₃	#461	1	2	yes	—
170	C₂²×C₇₃	#465	1	2	yes	—
171	C₂²×C₇₃	#471	1	2	yes	—
172	C₂²×C₇₃	C₂×#121	1	1	yes	characteristic
173	C₂×D₇₃	—	2	4	no	—
174	C₂×D₇₃	—	4	8	no	—
175	C₂×D₇₃	—	4	2	no	—
176	C₂×D₇₃	—	4	2	no	—
177	C₄×C₇₃	—	2	1	no	—
178	C₄×C₇₃	—	2	1	no	—
179	C₄×C₇₃	—	8	2	no	—
180	C₄×C₇₃	#461	1	2	yes	—
181	C₄×C₇₃	#465	1	2	yes	—
182	C₄×C₇₃	C₂²×SD₁₆	1	1	yes	characteristic
183	C₄×C₇₃	C₂²×SD₁₆	1	1	yes	characteristic
184	C₄×C₇₃	C₂×#135	1	1	yes	characteristic
185	C₄×C₇₃	C₂×#137	1	1	yes	characteristic
186	Dic₇₃	—	4	2	no	—
187	Dic₇₃	—	8	2	no	—
188	C₂×#1671	—	73	1	no	maximal, Sylow
189	C₂×C₇₃	#1671	1	4	yes	complemented
190	C₂×C₇₃	C₂×#461	1	1	yes	characteristic
191	C₂×C₇₃	C₂×#465	1	1	yes	characteristic
192	C₂×C₇₃	C₂×#471	1	1	yes	characteristic
193	D₇₃	—	4	4	no	—
194	C₂×#498	—	73	1	no	—
195	C₂×#476	—	73	1	no	—
196	C₂×#461	—	73	1	no	—
197	C₂×C₈×D₄	—	73	1	no	—
198	#1671	—	73	8	no	—
199	C₂×#334	—	73	1	no	—
200	C₂×#333	—	73	1	no	—
201	C₂×#332	D₇₃	1	1	yes	characteristic, complemented
202	C₇₃	C₂×#1671	1	1	yes	characteristic, complemented, Sylow
203	C₂²×SD₁₆	—	146	1	no	—
204	C₂³×C₈	—	146	1	no	—
205	C₂×C₄⋊Q₈	—	73	1	no	—
206	C₂×C₄⋊Q₈	C₂×D₇₃	1	1	yes	characteristic
207	C₂×C₄×D₄	—	73	1	no	—
208	#498	—	73	4	no	—
209	#498	—	146	2	no	—
210	#476	—	73	4	no	—
211	#476	—	146	2	no	—
212	#461	—	73	4	no	—
213	#461	—	146	2	no	—
214	C₈×D₄	—	73	4	no	—
215	C₈×D₄	—	146	2	no	—
216	C₂×#106	—	146	1	no	—
217	C₂×#106	—	146	1	no	—
218	C₂×#106	—	146	1	no	—
219	C₂×C₄⋊C₈	—	73	1	no	—
220	C₂×C₄⋊C₈	C₂×D₇₃	1	1	yes	characteristic
221	C₂×C₄⋊C₈	C₂×D₇₃	1	1	yes	characteristic
222	C₂×#103	—	146	1	no	—
223	C₂×#102	—	146	1	no	—
224	C₂×C₂²⋊C₈	—	146	1	no	—
225	C₂×C₄×C₈	—	73	1	no	—
226	#334	—	73	4	no	—
227	#333	—	73	4	no	—
228	#332	C₂×D₇₃	1	4	yes	complemented
229	C₂²×Q₈	—	2	1	no	—
230	C₂²×Q₈	—	146	1	no	—
231	C₂²×D₄	—	73	1	no	—
232	C₂³×C₄	—	146	1	no	—
233	C₂×SD₁₆	—	292	4	no	—
234	C₂×SD₁₆	—	146	4	no	—
235	C₂²×C₈	—	2	1	no	—
236	C₂²×C₈	—	2	1	no	—
237	C₂²×C₈	—	292	4	no	—
238	C₂²×C₈	—	73	1	no	—
239	C₂²×C₈	—	73	1	no	—
240	C₂²×C₈	—	146	4	no	—
241	C₂²×C₈	—	146	1	no	—
242	C₄⋊Q₈	—	2	1	no	—
243	C₄⋊Q₈	—	2	1	no	—
244	C₄⋊Q₈	—	2	1	no	—
245	C₄⋊Q₈	—	73	2	no	—
246	C₄⋊Q₈	—	146	1	no	—
247	C₄⋊Q₈	—	146	1	no	—
248	C₄⋊Q₈	—	146	1	no	—
249	C₄⋊Q₈	C₂²×D₇₃	1	2	yes	—
250	C₄×D₄	—	73	2	no	—
251	C₄×D₄	—	73	2	no	—
252	C₄×D₄	—	146	1	no	—
253	C₄×D₄	—	146	1	no	—
254	C₂×C₄⋊C₄	—	2	1	no	—
255	C₂×C₄⋊C₄	—	2	1	no	—
256	C₂×C₄⋊C₄	—	73	1	no	—
257	C₂×C₄⋊C₄	—	146	1	no	—
258	C₂×C₄⋊C₄	—	146	1	no	—
259	C₂×C₂²⋊C₄	—	146	1	no	—
260	C₂×C₄²	C₂²×D₇₃	1	1	yes	characteristic
261	#106	—	146	4	no	—
262	#106	—	146	4	no	—
263	#106	—	146	4	no	—
264	C₄⋊C₈	—	2	1	no	—
265	C₄⋊C₈	—	2	1	no	—
266	C₄⋊C₈	—	73	2	no	—
267	C₄⋊C₈	—	146	1	no	—
268	C₄⋊C₈	C₂²×D₇₃	1	2	yes	—
269	C₄⋊C₈	C₂²×D₇₃	1	2	yes	—
270	#103	—	146	4	no	—
271	#102	—	146	4	no	—
272	C₂²⋊C₈	—	146	4	no	—
273	C₄×C₈	—	73	2	no	—
274	C₄×C₈	—	73	2	no	—
275	C₂⁴	—	146	1	no	—
276	C₂×Q₈	—	2	2	no	—
277	C₂×Q₈	—	4	1	no	—
278	C₂×Q₈	—	4	2	no	—
279	C₂×Q₈	—	4	2	no	—
280	C₂×Q₈	—	292	1	no	—
281	C₂×Q₈	—	584	2	no	—
282	C₂×Q₈	—	146	2	no	—
283	C₂×D₄	—	73	4	no	—
284	C₂×D₄	—	73	2	no	—
285	C₂×D₄	—	146	1	no	—
286	C₂×D₄	—	146	2	no	—
287	C₂²×C₄	—	4	1	no	—
288	C₂²×C₄	—	292	2	no	—
289	C₂²×C₄	—	292	2	no	—
290	C₂²×C₄	—	292	1	no	—
291	C₂²×C₄	—	146	1	no	—
292	C₂²×C₄	—	146	2	no	—
293	C₂²×C₄	—	146	2	no	—
294	C₂²×C₄	D₄×D₇₃	1	1	yes	characteristic
295	C₂²×C₄	D₄×D₇₃	1	1	yes	characteristic
296	C₂²×C₄	#11924028	1	1	yes	characteristic
297	SD₁₆	—	292	8	no	—
298	C₂×C₈	—	2	2	no	—
299	C₂×C₈	—	2	2	no	—
300	C₂×C₈	—	4	2	no	—
301	C₂×C₈	—	4	2	no	—
302	C₂×C₈	—	292	8	no	—
303	C₂×C₈	—	292	2	no	—
304	C₂×C₈	—	73	4	no	—
305	C₂×C₈	—	73	2	no	—
306	C₂×C₈	—	73	2	no	—
307	C₂×C₈	—	146	2	no	—
308	C₂×C₈	—	146	2	no	—
309	C₄⋊C₄	—	2	2	no	—
310	C₄⋊C₄	—	2	2	no	—
311	C₄⋊C₄	—	4	1	no	—
312	C₄⋊C₄	—	4	1	no	—
313	C₄⋊C₄	—	292	1	no	—
314	C₄⋊C₄	—	292	1	no	—
315	C₄⋊C₄	—	73	2	no	—
316	C₄⋊C₄	—	146	2	no	—
317	C₄⋊C₄	—	146	2	no	—
318	C₄⋊C₄	—	146	1	no	—
319	C₂²⋊C₄	—	146	2	no	—
320	C₂²⋊C₄	—	146	2	no	—
321	C₄²	D₄×D₇₃	1	1	yes	characteristic
322	C₄²	D₄×D₇₃	1	1	yes	characteristic
323	C₄²	#11924028	1	1	yes	characteristic
324	C₄²	C₂³×D₇₃	1	1	yes	characteristic
325	C₂³	—	292	2	no	—
326	C₂³	—	292	2	no	—
327	C₂³	—	146	4	no	—
328	C₂³	—	146	2	no	—
329	C₂³	#14556564	1	1	yes	characteristic
330	Q₈	—	4	2	no	—
331	Q₈	—	4	2	no	—
332	Q₈	—	8	2	no	—
333	Q₈	—	584	2	no	—
334	Q₈	—	584	2	no	—
335	D₄	—	73	4	no	—
336	D₄	—	146	4	no	—
337	D₄	—	146	2	no	—
338	C₂×C₄	—	2	2	no	—
339	C₂×C₄	—	4	2	no	—
340	C₂×C₄	—	8	2	no	—
341	C₂×C₄	—	292	2	no	—
342	C₂×C₄	—	292	2	no	—
343	C₂×C₄	—	292	4	no	—
344	C₂×C₄	—	292	2	no	—
345	C₂×C₄	—	292	2	no	—
346	C₂×C₄	—	584	2	no	—
347	C₂×C₄	—	146	2	no	—
348	C₂×C₄	#14556564	1	1	yes	characteristic
349	C₂×C₄	#14556564	1	1	yes	characteristic
350	C₂×C₄	#14556564	1	1	yes	characteristic
351	C₂×C₄	SD₁₆×D₇₃	1	2	yes	—
352	C₂×C₄	SD₁₆×D₇₃	1	2	yes	—
353	C₂×C₄	#14556610	1	2	yes	—
354	C₂×C₄	#14556623	1	2	yes	—
355	C₂×C₄	C₂×D₄×D₇₃	1	1	yes	characteristic
356	C₂×C₄	C₂×D₄×D₇₃	1	1	yes	characteristic
357	C₂×C₄	C₂×#11924028	1	1	yes	characteristic
358	C₈	—	4	2	no	—
359	C₈	—	4	2	no	—
360	C₈	—	292	2	no	—
361	C₈	—	73	4	no	—
362	C₈	—	146	2	no	—
363	C₂²	—	292	8	no	—
364	C₂²	—	292	2	no	—
365	C₂²	—	292	2	no	—
366	C₂²	—	146	4	no	—
367	C₂²	#17988791	1	2	yes	—
368	C₂²	#17988815	1	2	yes	—
369	C₂²	#17988853	1	2	yes	—
370	C₂²	C₂×#14556564	1	1	yes	characteristic
371	C₄	—	2	1	no	—
372	C₄	—	2	1	no	—
373	C₄	—	8	2	no	—
374	C₄	—	292	2	no	—
375	C₄	—	584	2	no	—
376	C₄	#17988791	1	2	yes	—
377	C₄	#17988815	1	2	yes	—
378	C₄	C₂×SD₁₆×D₇₃	1	1	yes	characteristic
379	C₄	C₂×SD₁₆×D₇₃	1	1	yes	characteristic
380	C₄	C₂×#14556610	1	1	yes	characteristic
381	C₄	C₂×#14556623	1	1	yes	characteristic
382	C₂	—	292	4	no	—
383	C₂	#22609305	1	4	yes	direct product factor
384	C₂	C₂×#17988791	1	1	yes	characteristic
385	C₂	C₂×#17988815	1	1	yes	characteristic
386	C₂	C₂×#17988853	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#14556564
commutator, cosocle	C₄²×C₇₃	C₂⁴
Frattini	C₄²	C₂³×D₇₃
Fitting	C₂×#332×C₇₃	C₂
socle	C₂³×C₇₃	#121
prime radical	C₂²×D₂₉₂	D₄
hypercenter	C₂×#332	D₇₃
hypocenter	C₇₃	C₂×#1671
LCS	G ▷ C₄²×C₇₃ ▷ C₂²×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#332

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
1. C₂×#17988983	C₂	semidirect
5. #22609305	C₂	direct (N × H)
6. C₂×#17988337	C₂	semidirect
7. C₂×#17988335	C₂	semidirect
8. C₂×#332×C₇₃	C₂	semidirect
14. #17988983	C₂²	semidirect
29. #17988337	C₂²	semidirect
30. #17988335	C₂²	semidirect
34. #332×C₇₃	C₂²	semidirect
99. C₂×#10596831	SD₁₆	semidirect
168. #10596831	C₂×SD₁₆	semidirect
189. C₂×C₇₃	#1671	semidirect
201. C₂×#332	D₇₃	semidirect
202. C₇₃	C₂×#1671	semidirect
228. #332	C₂×D₇₃	semidirect
383. C₂	#22609305	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂×#17988983	2
C₂×#17988878	2
C₂×#17988789	2
C₂×#17988652	2
#22609305	2
C₂×#17988337	2
C₂×#17988335	2
C₂×#332×C₇₃	2
C₂×#1671	73

Subgroups — C2×#22609305 #33068232

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#22609305 #33068232